Approximation of the stability number of a graph via copositive programming

SIAM Journal on Optimization

Volumn 12, Issue 4, 2002, Pages 875-892

  De Klerk, E ^a   Pasechnik, D V ^a  

^a DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Approximation algorithms; Copositive cone; Lifting; Semidefinite programming; Stability number

Indexed keywords

EID: 0036433723     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623401383248     Document Type: Article

References (26)

1. N. Alon, J.H. Spencer, and P. Erdös, The Probabilistic Method Wiley, New York, 1992.
2. M.F. Anjos and H. Wolkowlcz, A strengthened SDP relaxation via a second lifting for the max-cut problem, Discrete Appl. Math., to appear.
3. I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo, The maximum clique problem, in Handbook of Combinatorial Optimization, Suppl. Vol. A, D.-Z. Du and P.M. Pardalos, eds., Kluwer, Dordrecht, The Netherlands, 1999, pp. 1-74.
4. I.M. Bomze, M. Dür, E. De Klerk, C. Roos, A. Quist, and T. Terlaky, On copositive programming and standard quadratic optimization problems, J. Global Optim., 18 (2000), pp. 301-320.
5. R. Boppana and M.M. Halldórsson, Approximating maximum independent sets by excluding subgraphs, BIT, 32 (1992), pp. 180-196.
6. H.S.M. Coxeter, Introduction to Geometry, Wiley, New York, 1969.
7. M.X. Goemans and L. Tunçel, When Does the Positive Semidefiniteness Constraint Help in Lifting Procedures, manuscript, Department of Mathematics, MIT, Cambridge, MA, 2000.
8. M.X. Goemans and D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), pp. 1115-1145.
9. M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988.
10. 1-ε|, Acta Math., 182 (1999), pp. 105-142.
11. M. Laurent, Tighter linear and semidefinite relaxations for max-cut based on the Lovász-Schrijver lift-and-project procedure, SIAM J. Optim., 12 (2001), pp. 345-375.
12. M. Laurent, A Comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre Relaxations for 0-1 Programming, Technical report PNA-R0108, CWI, Amsterdam, 2001.
13. J.B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796-817.
14. J.B. Lasserre, An explicit exact SDP relaxation for nonlinear 0-1 programs, in Proceedings of the 8th International Integer Programming and Combinatorial Optimization Conference, Lent. Notes in Comput. Sci. 2081, Springer, Berlin, 2001, pp. 293-303.
15. L. Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory, 25 (1979), pp. 1-7.
16. L. Lovász and A. Schrijver, Cones of matrices and set-functions and 0-1 optimization, SIAM J. Optim., 1 (1991), pp. 166-190.
17. T.S. Motzkin and E.G. Straus, Maxima for graphs and a new proof of a theorem of Turán, Canadian J. Math., 17 (1965), pp. 533-540.
18. K.G. Murty and S.N. Kabadi, Some NP-complete problems in quadratic and linear programming, Math. Programming, 39 (1987), pp. 117-129.
19. Yu. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Stud. Appl. Math. 13, SIAM, Philadelphia, 1994.
20. P.A. Parrilo, Structured Semidefinite Programs and Semi-algebraic Geometry Methods in Robustness and Optimization, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 2000. Available online at: http://www.cds.caltech.edu/̃pablo/.
21. G. Pólya, Über positive Darstellung von Polynomen, Vierteljschr. Naturforsch. Ges. Zürich, 73 (1928), pp. 141-145; Collected Papers, Vol. 2, MIT Press, Cambridge, MA, London, 1974, pp. 309-313.
22. G. Pólya, Über positive Darstellung von Polynomen, Vierteljschr. Naturforsch. Ges. Zürich, 73 (1928), pp. 141-145; Collected Papers, Vol. 2, MIT Press, Cambridge, MA, London, 1974, pp. 309-313.
23. V. Powers and B. Reznick, A new bound for Pólya's theorem with applications to polynomials positive on polyhedra, J. Pure Appl. Algebra, 164 (2001), pp. 221-229.
24. A.J. Quist, E. De Klerk, C. Roos, and T. Terlaky, Copositive relaxation for general quadratic programming, Optim. Methods Softw., 9 (1998), pp. 185-209.
25. J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, SIAM, Philadelphia, 2001.
26. A. Schrijver, A comparison of the Delsarte and Lovász bounds, IEEE Trans. Inform. Theory, 25 (1979), pp. 425-429.